# Concept of Differentiation

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## Differentiation

In maths, Differentiation Class 11 is one of the most important topics both academically and in terms of marks weightage. The concept of differentiation refers to the method of finding the derivative of a function. It is the process of determining the rate of change in function on the basis of its variables. The opposite of differentiation is known as anti-differentiation. Suppose, we have two variables x and y. Then the rate of change of x with respect to y and is denoted as dy/dx. The general expression of derivative of a function is f’(x)= dy/dx where y= f(x) is any function.

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### What is Differentiation in Mathematics?

Differentiation in mathematics is defined as a derivative of a function in terms of an independent variable.

Let f(x) be a function of x and y be another variable.

Here, the rate of change of y per unit change in x is denoted by dy/dx.

If the function f(x) goes through an infinitesimal change of h near to any point x, the function is defined as,

Lim f(x + h) - f(x)/ h

H tends to 0.

### What is Differentiation in Physics?

Differentiation in physics is the same as differentiation in mathematics. The concepts from differentiation in maths are used in physics too.

### Some Important Formulas in Differentiation

Some important Differentiation Formula Class 11 are given below. We have to consider f(x) as a function and f’(x) as the derivative of the function:

1. If f(x) = tan(x) then f’(x) = sec2x.

2. If f(x) = cos(x) then f’(x) = -sin x.

3. If f(x) = sin(x) then f’(x) = cos x.

4. If f(x) = In(x) then f’(x) = 1/x.

5. If f(x) = ex then f’(x) = ex.

6. If f(x) = xn then f’(x) = nxn-1 where n is any fraction or integer.

7. If f(x) = k then f’(x) = 0 and here k is a constant.

### Rules of Differentiation

The main differentiation rules that need to be followed are given below:

1. Product Rule

2. Sum and Difference Rule

3. Chain Rule

4. Quotient Rule

Product Rule - According to the product rule, if the function f(x) is the product of two functions suppose a(x) and b(x), then the derivative of that function is:

If f(x) = a(x) * b(x) then,

f’(x) = a’(x) * b(x) + a(x) * b’(x)

Sum and Difference Rule - According to the sum and difference rule, if the function f(x) is the sum or difference of two functions suppose a(x) and b(x), then the derivative of the function is as follows:

If f(x) = a(x) + b(x) then,

f’(x) = a’(x)+ b’(x)

If f(x) = a(x) - b(x) then,

f’(x) = a’(x) - b’(x)

Chain Rule - If a function y= f(x) = g(u) and if u = h(x), then according to the chain rule for differentiation,

dy/dx = dy/du * du/dx

This rule is very important in the method of substitution during differentiation of composite functions.

Quotient Rule - If the function f(x) is the quotient of two functions i.e. [ a(x)/b(x)], then according to quotient rule, the derivative of the function is as follows:

If f(x) = a(x)/b(x) then,

f’(x) = a’(x) * b(x) - a(x) * b’(x) / (b(x))2

### Solved Example

1. Differentiate f(x) = 9x3 - 6x + 5 with respect to x.

Solution - Here, f(x) = 9x3 - 6x + 5

Differentiating both sides w.r.t. x we get,

f’(x) =(3)(9)x2 - 6

f’(x) = 27x2 - 6

FAQ (Frequently Asked Questions)

Q1. What is Differentiation? Give Some Real-Life Differentiation Examples.

Ans. Differentiation meaning in maths is defined as a derivative of a function in terms of an independent variable. In mathematics, it is used to measure the function per unit change in the independent variable. The opposite of differentiation is known as anti-differentiation.

We can find the rate of change of one quantity with respect to another with the help of differentiation. It is useful in many ways.

Some of the real-life differentiation examples are:

• Acceleration.

• It helps us to derive tangent and normal to a curve.

• It helps to calculate the highest or lowest points of a curve in a graph.

Q2. What are the Differentiation Formulae Class 11 Physics?

Ans. The differentiation formulas in class 11 physics are as follows. We have to consider f(x) as a function and f’(x) as the derivative of the function:

If f(x) = tan(x) then f’(x) = sec2x.

If f(x) = cos(x) then f’(x) = -sin x.

If f(x) = sin(x) then f’(x) = cos x.

If f(x) = In(x) then f’(x) = 1/x.

If f(x) = ex then f’(x) = ex.

If f(x) = xn then f’(x) = nxn-1 where n is any fraction or integer.

If f(x) = k then f’(x) = 0 and here k is a constant.